Tracking the influence of a very close secondary star on planetary growth and evolution

ABSTRACT

This work investigated the dynamics of planets in binary systems and provided insights into the stability and evolution of these systems. We explored the influence of a nearby secondary star on planetary growth and evolution, focusing on S-type configurations. We tracked the orbits of the planets and analysed their stability over long time-scales, considering various parameters such as mass, eccentricity, and inclination. Our results show that the presence of a secondary star can significantly impact the growth and evolution of planets, leading to changes in their orbits and potential ejection from the system, however, it was possible to identify stable planets even in systems experiencing multiple disturbances. One of the most significant results of the work was the analysis of the increased material in the disc near the primary star, which contributes to planet growth, driven by the density spirals influenced by the binary star. These findings have important implications for the search for habitable exoplanets and emphasize the need for further studies of planetary systems in binary star environments.

1 INTRODUCTION

There is a growing number of multiple-star systems being discovered. Among them, 154 binary-star systems are confirmed, and within these systems, 217 planets have been detected so far (Schwarz et al. 2016). Additionally, planets in systems with three or more stars have also been recorded. For instance, the first planet found in a multiple-star system was discovered in a triple system, 16 Cyg (Cochran et al. 1997). The closest known exoplanet is also part of a triple system, Proxima Cen (Anglada-Escudé et al. 2016). In this work, we will focus on binary systems in Type-S configuration.

In multiple-star systems, a planet in an S-type configuration is orbiting only one of the stars while the other stellar bodies perturb the orbit of the planet. If the stars are close to each other, this proximity results in several interactions that might alter the dynamics of a planetary system (Rabl & Dvorak 1988).

The study by Thebault & Haghighipour (2015) investigates planet formation in binary systems, emphasizing the challenges posed by a companion star, which influences key stages like protoplanetary disc formation and planetesimal accretion. Simulations show that protoplanets may experience dynamic excitation, affecting their formation and stability, particularly in habitable zones. Despite advancements, the study identifies unresolved issues needing further research, calling for detailed simulations to understand dynamic interactions in these environments. Similarly, Silsbee & Rafikov (2021) focus on the destructive collisions of planetesimals in tight binary systems, introducing a novel numerical framework for coagulation-fragmentation calculations. Their global simulations identify critical parameters for planetesimal growth, particularly the alignment of the protoplanetary disc with the binary orbit, which fosters stable zones for accumulation. The findings underscore the importance of tranquil regions for growth and support the instability of flow as a planetesimal formation mechanism, ultimately enhancing theories of planetary formation around both binary and single stars.

Following this, Zagaria et al. (2023) review the evolution of protoplanetary discs in binary systems, focusing on dust dynamics and its implications for planet formation. They note that companion stars typically lead to smaller, less massive discs with shorter lifespans compared to single-star systems. Their analysis of recent numerical simulations reveals that tidal truncation limits the material available for planet formation and inhibits dust coagulation, complicating the emergence of planetary embryos. Additionally, the intricate dust dynamics highlight the need for advanced models to fully understand these interactions in binary environments.

The data from the Kepler mission suggest that the occurrence rate of planets in binary stars with orbital separation |$\lt 47$| au is only one-third of that in wider binaries or single-star systems. This indicates that planets in binary systems are less common than those in wider binary systems or single-star systems (Kraus et al. 2016).

Systems such as HD 41004, HD 196885, and HD 120136 (Raghavan et al. 2010; Satyal, Quarles & Hinse 2013) exhibit a secondary star very close to the primary and a giant planet orbiting the primary star. Tables 1 and 2 present some parameters of these systems. In the systems presented in Table 1, the secondary stars in the HD 196885 and HD 41004 systems have similar values for the semimajor axis and eccentricity. However, the HD 120136 system is an odd case for binary-star systems. While the semimajor axis of the secondary star is about 45 au, its high eccentricity (approximately 0.91) allows the secondary star to approach its primary companion at around 4 au on its pericentre.

Another intriguing case is the HD 42936 system, a binary star system in the Canis Major constellation. It consists of two stars similar to the Sun: HD 42936, a yellow dwarf star with approximately 0.9 solar masses (⁠|${\rm M}_{\odot }$|⁠); and HD 42936 B, with about 0.8 |${\rm M}_{\odot }$|⁠. In 2005, the planet HD 42936 b was discovered in the system using the radial velocity method. HD 42936 b is a gas giant planet with a minimum mass of around 1.7 Jupiter’s masses, and it is located at approximately 1.2 au from the primary star, placing it within the habitable zone of HD 42936 (Simonetti et al. 2020).

Moe & Kratter (2021) investigate the influence of binary stars at different separations on planet formation, emphasizing that close binaries, especially those with separations less than 1 au, suppress the formation of S-type planets. This suppression decreases as the separation increases but remains significant at smaller distances. Binaries at 10 au host close planets at 15īper cent of the rate observed in single stars, while binaries with separations greater than 200 au have an almost insignificant effect. For this analysis, the authors compiled data from radial velocity surveys and high-resolution images, making corrections for bias and incompleteness. They measured the binary fraction and period distribution concerning stellar mass, using statistical analyses to relate binary separation to planet occurrence, as well as comparing different detection methods. While the study relies on observational data, we propose an alternative approach through simulations that explore the hydrodynamic evolution of protoplanets in various binary system environments.

It is well known that the formation of planets in compact binary systems is strongly influenced by stellar parameters, as the protoplanetary disc and its inner particles are subject to the gravitational influence of the secondary star. Thus, not only gas drag can alter the evolution of embedded protoplanets, as observed in systems with a single star or distant companions, but also the gravitational influence of the disc and the secondary star (Gyergyovits et al. 2014).

The planets in the HD 196885 and HD 41004 systems, whose characteristics are presented in Table 2, exhibit significant similarities to the previously studied planet. The HD 120136 system stands out from the other three systems. As observed in Table 2, the planet’s characteristics indicate a hot-Jupiter in a nearly circular orbit very close to the primary star. In this case, the fact that the secondary star approaches within 4 au might have contributed to the formation or migration of the planet to this region due to disc truncation and material accumulation in the innermost part of the disc. The majority of planets with measured masses in multiple stellar systems are more massive than Neptune. However, there is a high likelihood of many low-mass planets yet to be discovered in these systems.

The secondary star can influence the formation of a planet in a circumprimary system. Moreover, the presence of a close companion might lead to significant disturbances in the disc, including high eccentricity and retrograde precession (Jordan et al. 2021). These factors make the study of protoplanetary disc dynamics in close binaries an important challenge to better understand the planetary formation in these systems.

Regály et al. (2011) also aimed to model the evolution of the disc’s eccentricity in a binary system under the gravitational perturbation of the companion using 2D hydrodynamic simulations. They found that the disc’s eccentricity plays a major role in planet formation in binary systems. Hydrodynamic simulations revealed that the disc’s eccentricity can be excited and damped by the gravitational perturbation of the companion star, and it can significantly affect the migration and capture of planetesimals. However, the formation of planets in these scenarios was not explored in the study.

In Camargo, Kley & Winter (2023), the authors studied the conditions for the formation of the planet |$\gamma$| Cephei b. The planet formed within a gas disc perturbed by the presence of the secondary star located at a distance of 20 au from the primary star. In the present study, we generalize the work of Camargo et al. (2023) by investigating the influences of the different secondary stars on the formation of planets similar to |$\gamma$| Cephei b. To do so, we explore different ranges of mass and semimajor axis for the secondary star. The test stars are simulated in conjunction with the primary star, the gas disc, and the planet in the accretion phase.

In summary, our primary objective is to explore how the mass and semimajor axis of the secondary star influence the formation of planets in compact binary systems using hydrodynamics simulations. In Section 2, we delineate our methodology and present the parameters of the systems investigated. In Section 3, we discuss the results of the models of disc truncation. Then, in Section 4, we analyse the planet growth and evolution in each explored model. The summary of our results and conclusions are presented in Section 5.

2 MODEL SETUP

To properly study the evolution of planets in close binary systems, we approach the problem using numerical hydrodynamic simulations. The fargo 2D (Fast Advection in Rotating Gaseous Object) code (Masset 2000), is employed for solving the hydrodynamic aspects of these simulations. This code, which combines a simple and fast 2D hydrodynamical-hybrid approach with N-body simulations, is dedicated to modelling planetary–disc tidal interactions and planet–planet gravitational interactions. It is based on the Navier–Stokes equations and uses a solver similar to the one found in the zeus code (Stone & Norman 1992). Additionally, the positions of the celestial bodies are integrated using a fifth-order Runge–Kutta method, which is already implemented in fargo 2D (Masset 2000). It is important to highlight that our study use a gas disc without the inclusion of dust.

The gravitational influences of the central mass, embedded protoplanet, and secondary star were considered on the disc. To prevent numerical issues near the planet, where the planet’s gravitational potential diverges, a smoothing length term, denoted as |$r_{\mathrm{ SM}}$|⁠, is introduced. The smoothing length depends on approximately 1/5 of the size of the Roche lobe of the planet and the reduced mass of the system. The exact value of |$r_{\mathrm{ SM}}$| is not so critical, as long as it is substantially smaller than the Roche radius of the planet (Kley 1999). The simulations used polar coordinates centred on the primary star without accounting for self-gravity or magnetism.

The planetary accretion region is defined with respect to its Hill radius (⁠|$R_\mathrm{ H}$|⁠). Utilizing the parameters outlined by Kley (1999), this accretion zone is partitioned into three subzones, each corresponding to distinct accretion rates. Specifically, the parameters |$f_1 = 0.75$| and |$f_2 = 0.45$| delineate the proportions of the planet’s Hill radius employed for these accretion zones. The region enclosed by |$f_1 R_\mathrm{ H}$| represents the largest zone capable of gas accretion. Within this zone, the amount of gas that can be accreted is restricted to one-third of the total gas content within the area over a given unit of time. Conversely, the smaller zone defined by |$f_2 R_\mathrm{ H}$| is constrained to the remaining two-thirds of the gas in the area. Consequently, this implies that cells within |$r \lt f_2 R_\mathrm{ H}$| of the planet would undergo accretion twice in the span of a single time unit.

In our research, we use a modified version of fargo, as described in Müller & Kley (2012). The fargo code was adapted to meet the specific demands of studying disc dynamics in binary systems. The operator splitting method was employed to update velocities with source terms such as pressure gradients, viscous accelerations, and centrifugal forces. A first-order integrator was used to update velocities with source terms. The position of the secondary star is calculated using a fifth-order Runge–Kutta algorithm. These modifications to the original code enable researchers to conduct detailed simulations of protoplanetary disc dynamics in binary systems, considering aspects such as disc eccentricity, energy transfer, eccentricity growth rate, and interactions with tidal torques. These adjustments were crucial for investigating how disc dynamics are influenced by the presence of a secondary star in binary systems and its impact on planetary formation.

According to Müller & Kley (2013), the relationship between torque and the eccentricity of the disc is described by the following equation:

where |${\bf L}$| and |$\Sigma$| represent the disc’s angular momentum and surface density, respectively, r is the radial distance, |$\Omega$| is the angular velocity, |${\bf v}$| is the gas velocity in the disc, and |$\mathrm{ d}A$| is the area element. The eccentricity of the disc enters indirectly through the distribution of mass and velocity in the protoplanetary disc.

This equation describes the relationship between the torque exerted on the disc and the evolution of its eccentricity, taking into account the mass distribution and velocity in the protoplanetary disc near the protoplanet. The eccentricity of the disc can influence how the torque acts in the system, affecting the dynamics and evolution of the disc around a forming planet.

The disc viscosity is modelled using the alpha-disc formalism proposed by Shakura & Sunyaev (1973), allowing for the description of general disc properties with a limited number of parameters (Table 3).

To study the influences of the secondary star on the formation of a protoplanet around the primary star, we explore a range of values for the semimajor axis and mass of the secondary star. Given the parameters shown in Table 3 for the gas disc and the parameters shown in Table 4 for the secondary star, in all cases, the secondary star has an eccentricity equal to 0.4. We will assess the evolution of the gas disc, focusing particularly on the protoplanet immersed in this disc.

In all cases, we assume that the protoplanet starts in a circular orbit with a semimajor axis of 3 au. The protoplanet initially has 10 per cent of the mass of Jupiter (⁠|$M_{\mathrm{ Jup}}$|⁠) and is accreting mass from the disc after 1000 yr of integration, and the planet’s orbit evolves in time. In all simulations, the primary star’s mass was set to |$1.4\,{\rm M}_{\odot }$|⁠, based on parameters from the |$\gamma$| Cephei system studied in Camargo et al. (2023).

As shown in Camargo et al. (2023), the differences between the isothermal and radiative discs are neglectable. On the other hand, the computational time required for the radiative discs is much greater than that for the isothermal discs. In this case, we choose to work with isothermal discs for all cases presented here.

Since our study is based on the extensively researched |$\gamma$| Cephei system, where the separation between the stars is 20 au, we selected both smaller and larger semimajor axis values to track the influence of the secondary star on planetary formation. Our goal was to generalize the |$\gamma$| Cephei system model. A total of 13 simulations were conducted: one without a binary star (index SB) and 12 with a binary star (index B). We chose three different semimajor axes for the binary stars: 20 au, similar to the |$\gamma$| Cephei system, 10 au for closer binaries, and 30 au for wider binaries. To further examine the influence of the secondary star’s mass, we set its mass at 0.1, 0.4, 1.0, and 1.4 solar masses.

For the cases at 10 and 20 au, the systems were integrated for 3000 yr, while at 30 au, the integration was performed for 7000 yr. In all cases, the initial 1000 yr were dedicated to achieving equilibrium in the disc in response to the system’s forces. This phase includes embedding a non-accreting protoplanet into the gas disc. To sensitize the disc to the exerted forces, an initial simulation was imperative. In this initial millennium, the protoplanet was influenced only by the secondary star, while the disc responded to the presence of both the protoplanet and the stars within the system. Consequently, these interactions led to shifts in the disc’s gas mass and alterations in its gas density distribution.

Figs 1 and 2 show the gas density distribution of the gas disc at time zero and after a thousand years, respectively, for the case without the binary companion. As one can see, even when the presence of the companion star is suppressed, the circumstellar disc presents spiral patterns. Spiral arms might occur due to different sources. In this case, the torques exerted by the protoplanet on the disc are responsible for the structures.

The overall torque is composed of the combination of the Lindblad and the corotation torques. The magnitude of this total torque inversely correlates with the square of the disc’s thickness or is in inverse proportion to the disc temperature, while it exhibits a linear relationship with the mass of the protoplanetary gas disc. The external disc situated beyond the protoplanet’s orbital path exerts a detrimental torque on the planet, whereas the internal disc imposes a beneficial torque. In the majority of disc models, the negative torque prevails, resulting in the protoplanet’s inward migration (Nelson, Papaloizou & Masset 2004).

Because of the diversity of masses and semimajor axes we consider for the companion star, the disc’s structure after the first thousand years will be different for each case. As we will see later, these structures are decisive for the protoplanet’s evolution.

3 DISC TRUNCATION

As discussed, the gas disc is crucial for planetary formation. In the case of compact binary systems, not only will the direct influence of the secondary star on the protoplanet affect its evolution, but also the disturbance in the gas disc, which will indirectly impact planet formation.

The amount of material available in the disc is essential in determining the protoplanet’s evolution. To understand the evolution of the disc’s mass, we present the data from the 13 performed cases. It is important to note that these simulations were conducted with open boundary conditions, meaning that the gas can freely leave the system. With the outflow of mass from the system, not all the lost material is accreted by the protoplanet.

In Figs 3 and 4, we have the set of plots related to the disc disturbed by a secondary star with a semimajor axis of 10 au after a thousand year.

In all cases, the star begins the simulation at its apocentre, in this case, at 14 au. We observe that in this set of simulations, the gas disc significantly reduces its radius, and a portion of the disc material is launched towards the primary star, leaving the disc with a higher average density than the initial state, but only within a radius of approximately 3 au.

In the context of a 10 au separation, the orbital period across our scenarios exhibits variability within the range of 19–26 yr. Similarly, for a separation of 20 au, the orbital period demonstrates variation spanning from 53 to 73 yr. Furthermore, in instances involving the separation of the secondary star at 30 au, the orbital period may vary between approximately 98 and 134 yr. Considering that, in the case of 10 au, the orbital period is significantly shorter compared to other scenarios, the secondary star quickly interferes with the disc.

The average gas density undergoes significant variation, with case B101 (⁠|$0.1\, {\rm M}_{\odot }$|⁠) showing the smallest modification and doubling the initial average gas density value. Case B114 (⁠|$1.4\, {\rm M}_{\odot }$|⁠) exhibits nearly 10 times the value of the average density by the end of the disc truncation.

In all cases, we observe that the disc reduces its radius to less than 3 au, where the protoplanet starts the simulations. Therefore, due to the scarcity of material in this scenario, planet formation is not possible in this model at an initial semimajor axis of 3 au.

In Fig. 5, one can see that when the secondary star has a semimajor axis of 10 au and an eccentricity of 0.4, there is a significant perturbation in the gas disc. In all cases, the disc loses a substantial amount of mass in the initial years of the simulation compared to the case without the binary star (SB). After the first passage of the star, the disc shows minimal material loss in the first 1000 yr.

The eccentricity of the disc is also heavily influenced by the secondary star, as depicted in Fig. 5 (upper panel). For the case of the companion star with a mass of |$1.0\, {\rm M}_{\odot }$| (B110) and |$1.4\,{\rm M}_{\odot }$| (B114), the gas disc eccentricities exceed 0.3 in the early moments of the simulation, whereas in the case without the secondary star (SB), the disc exhibits nearly circular behaviour.

The hydrodynamical simulations of the disc yield the gas velocity for each grid cell, allowing the calculation of an individual eccentricity assigned to each cell. The eccentricity of the entire disc is then estimated using the mass-weighted average of the eccentricities of all cells, as proposed by Kley & Nelson (2008). The global disc eccentricity |$e_{\text{disc}}$| is determined through a mass-weighted average over the entire disc, with the integrals evaluated by summation over all grid cells (Kley 1999), as shown in equation (2) (Müller & Kley 2012),

The eccentricity behaviour reveals at least two frequencies. One frequency has a shorter period caused by the orbital period of the secondary star, and another longer period frequency likely caused by secular perturbation.

Figs 6 and 7 show the density distribution after a thousand years for cases where the secondary star has a larger semimajor axis, 20 au. In these cases, the secondary star also initiates the simulation at its apocentre, 28 au. The star has its pericentre at 12 au, generating spiral arms and elongating the gas disc at each close approach. In this way, as the star reaches farther distances, the gas disc tends to circularize.

The gas disc also exhibits spirals due to the planet’s perturbation on the disc. Notably, in the case of B201 (⁠|$0.1\, {\rm M}_{\odot }$|⁠), there is a spiral connecting the outer and inner parts of the disc generated by the protoplanet at 3 au. In the case of B214 (⁠|$1.4\, {\rm M}_{\odot }$|⁠), the protoplanet appears to capture a portion of the material in a co-orbital region. The gas disc also presents an asymmetry in the gas distribution. Despite the appearance of disc circularization, we can observe a bulge of material, and the primary star is not at the centre of the disc. The reasons for these asymmetries in the disc will be explored in future studies.

At 1000 yr, the average gas density distribution is altered, increasing by 200 per cent for B201, 300 per cent for B204, 400  per cent for B210, and 800 per cent for the case of B214.

The first plot of Fig. 8 shows the disc mass loss over the initial thousand years for a secondary star with a larger semimajor axis of 20 au and for the case without the binary star. It is noticeable that for cases where the mass of the secondary star exceeds |$0.4\, {\rm M}_{\odot }$|⁠, the amount of mass removed sharply decreases at the beginning of the simulation and remains relatively consistent throughout the first thousand years. In the case of B201, a significant gas mass is lost in the initial simulation phase, although it is less than in other cases with more massive binary stars. After the initial years, the lost gas mass follows a smoother trend. The SB case (without a binary star) maintains a gentle curve of mass loss from the start of the integration.

In the second plot of Fig. 8, we can see that the eccentricity can reach up to 0.25 in the cases with the more massive companion star. Additionally, there is a peak in the short-period perturbation due to the secondary star’s perihelion spaced farther apart than in the previous case due to the longer period of the star.

From Figs 9 and 10, we can analyse the new distribution of the gas disc for cases with a binary star having a larger semimajor axis of 30 au. In this scenario, the apocentre of the companion star is at 42 au, and the pericentre is at 18 au. This implies that the short-period perturbation will occur at wider intervals compared to the previous cases.

Figs 9 and 10 show a lower mass loss through the gas disc. Here, the formation of spiral arms due to the protoplanet and those caused by the perturbation of the secondary star becomes more evident. The secondary star even flattens the gas disc at its pericentre, but as it moves away, the disc tends to circularize again.

The new density distribution of the gas disc is also affected but with less intensity compared to the previous cases. Cases B301 (⁠|$0.1\, {\rm M}_{\odot }$|⁠) and B304 (⁠|$0.4\, {\rm M}_{\odot }$|⁠) do not experience a significant increase in gas disc density. For case B310 (⁠|$1.0\, {\rm M}_{\odot }$|⁠), the density value is double the initial value. Case B314 (⁠|$1.4 \,{\rm M}_{\odot }$|⁠) has the highest density value, triplicating its initial value.

First plot in Fig. 11 shows the influence of the secondary star on the gas disc mass. Cases B304, B310, and B314 exhibit a greater mass loss in this set, however, it is much lower compared to the simulation sets for other semimajor axes of the binary star. The second plot of Fig. 11 presents the eccentricity of the disc over the initial thousand years. The short-period perturbation due to the perihelion is noticeable, but, in general, the eccentricity values do not exceed 0.15.

We observed that with the approach of the secondary star, spiral arms form in the gas disc. As described in the work by Dong, Najita & Brittain (2018), these spiral arms are a result of mass transportation due to the secondary star and the protoplanet embedded within the gas disc.

We emphasize that the average gas disc density rises across all cases until the 1000-yr simulation mark. This rise stems from a notable influx of gas disc material towards the primary star, driven by the perturbations induced by both the stellar companion and the protoplanet. Because of this new structure, we utilize the truncated disc at 1000 yr as the initial disc for the beginning of the protoplanet’s material accretion process.

When examining the truncation of the disc, we observe that its radius appears to align with the values established by Papaloizou & Pringle (1977) for binary separations of |$a = 3 R_{\mathrm{ trunc}}$|⁠, where |$R_{\mathrm{ trunc}}$| denotes the truncation radius. However, we do not expect our results to exactly match those of Papaloizou & Pringle (1977), as we are using non-circular orbits and the stellar masses are not always equal. A key observation is that the presence of a secondary star – especially one with a smaller semimajor axis – results in a reduction of the gas disc radius. This finding is consistent with the results of Artymowicz & Lubow (1994).

4 PLANET GROWTH

As discussed earlier, the gas disc plays an important role in planetary formation. In the case of compact binary systems, not only will the direct influence of the secondary star interact with the protoplanet, but also the perturbation in the gas disc will indirectly impact planet formation.

The amount of material available in the disc is crucial in determining the protoplanet’s evolution. Thus, to comprehend the impacts of the disc’s mass over the formation of planets, we present the planetary formation data from the 13 cases studied.

From previous section, when the secondary star has a semimajor axis of 10 au, there is an accumulation of material at the centre of the disc. This accumulation significantly impacts planet formation due to the lack of material in the gas disc around 3 au, where the protoplanet is initially located. Fig. 12 shows the distribution of gas disc density when the protoplanet begins to accrete material from the gas disc. The dashed line represents the density distribution at time zero. One can see that in the region where the protoplanet starts (3 au), indicated by the grey vertical line, is scarce of gas when the secondary star is at 10 au, except for the case B101.

For the cases B104, B110, and B114 the amount of gas left in the disc remains constant throughout the simulation’s duration. This is a consequence of the earlier mass removal due to the disturbance caused by the secondary star. Also, since the protoplanet is ejected from the system, the disc tends to maintain a consistent mass. Only in the case B101, does the protoplanet remain within the system. This result shows that even in a system with perturbations from a very close star, a giant planet formation is possible, assuming the pre-existence of a giant planet core. In this scenario, the disc maintains a constant mass for the initial 1000 yr, and afterward, when the protoplanet begins accreting mass, the disc’s mass gradually decreases.

As the protoplanet interacts with the disc, spiral arms become more prominent. Also, as the protoplanet accreates the surrounding gas material, it creates a gap structure in the disc, leaving only the spiral arms connecting the forming planet with the remaining disc.

Although the protoplanet remained on the system in case B101, as shown in the bottom panel of Fig. 13, the protoplanet rapidly migrated inward and completed its formation much closer to the star than expected. In this scenario, greater stability only manifests after 1.5 thousand years, initially subjecting the disc to considerable perturbations.

In the cases of B104 and B114, the protoplanets experience dynamic instability and are ejected from the system. Conversely, B110 migrates toward the primary star and a collision takes place. Hence, for masses exceeding |$0.4 {\rm M}_{\odot }$|⁠, the 10 au approximation precludes the stability of the protoplanet in similar systems.

In the central plot of Fig. 13, we depict the evolution scenario of the protoplanet’s eccentricity for this set of simulations. As anticipated, it becomes evident that stability in this case is highly challenging due to the low value of the semimajor axis, coupled with the high eccentricity of the star, 0.4. However, even within this context, B101 manages to remain within the system with an eccentricity of approximately 0.15, as observed in Fig. 13. In the first 1000 yr, the disc influences the protoplanet, while the reverse does not hold true; consequently, the protoplanet only experiences forces relative to the stars in the system. Since the instability of cases B104, B110, and B114 arises early in the simulations, it is inconclusive whether the protoplanet feeling the disc’s influence from the integration’s outset could dampen or hinder its significant migration.

The top panel in Fig. 13 illustrates the growth of the protoplanet’s mass for the scenario with a star at 10 au. As observed, only the case of binary B101 remains within the system. In this B101 scenario, there is also a greater mass accretion compared to the case without the binary star. It is worth noting that, due to the disc’s truncation, the quantity of available material in the region where the protoplanet is initialized is meager. Migration inwardly allowed the incorporation of material from the disc. In the case without the secondary star (SB), the decline in available material within the gas disc occurs gradually and almost linearly.

Fig. 14 shows the gas disc density distribution at 1000 yr when the protoplanet begins to accrete material from the gas disc for cases where the secondary star has a semimajor axis of 20 au. The dashed line represents the density distribution at time zero. In the region where the protoplanet is located (3 au), indicated by the grey vertical line, all simulated cases (B201, B204, B210, and B214) exhibit a substantial amount of gas. However, it is worth noting that for the cases where the protoplanet was slightly further from its current position, the available gas quantity would no longer be as abundant.

In Fig. 15, we present the evolution of the semimajor axis, eccentricity, and mass of the protoplanet for cases involving a binary star with a semimajor axis of 20 au. In all four cases, namely B201, B204, B210, and B214, the secondary star influences the system, leading to faster protoplanet migration, as seen in the bottom plot in Fig. 15. When both stars have a mass of |$1.4\, {\rm M}_{\odot }$|⁠, the protoplanet undergoes significant migration, with its semimajor axis reaching approximately 2.3 au by the end of the simulation.

For the case B201, the influence of the secondary star is minimal (Fig. 15). Despite exhibiting peaks related to the binary’s perihelion, in the overall context, the behaviour of the semimajor axis evolution closely resembles the case without the binary.

An important point for the protoplanet’s stability is the analysis of eccentricity evolution. The central panel in Fig. 15 presents the evolution of the eccentricity for the case without the binary, SB, and the cases B201, B204, B210, and B214 for the secondary star with a semimajor axis of 20 au. We notice minimal variation in the SB case, while the other cases exhibit slightly higher peaks, reaching above 0.2 in the cases of B210 and B214. However, these two cases conclude the integration with considerably low values.

Short-period oscillations in the protoplanet’s eccentricity are also evident. This effect arises from close encounters of the secondary star with the protoplanet each time it passes through the pericentre. As expected, the amplitude of these oscillations increased with the secondary star’s mass.

For the case B204, the eccentricity gradually increases after 1500 yr, surpassing the value of 0.1. B201 exhibits minimal eccentricity variation.

The evolution of mass is another point of analysis in this study. Material accretion is essential for a planet’s evolution. The top panel in Fig. 15 presents results regarding the protoplanet’s mass evolution for the secondary star’s semimajor axis of 20 au. We found that in all cases with the binary, mass accretion is greater than in the absence of the companion star. For B204, B210, and B214, the growth exceeds the value of |$2 M_\mathrm{ J}$|⁠. In cases B210 and B214, this growth is more rapid between 1000 and 1500 yr, which may influence a transition from Type I to Type II migration after 1500 yr, given the material scarcity beyond this time interval.

We understand that, depending on the secondary star’s mass, the protoplanet’s material accretion rate is significantly higher when compared to the same system without a companion star or with a small mass value.

Fig. 16 shows the gas disc density distribution at 1000 yr, when the protoplanet begins to accrete material from the gas disc, for cases where the secondary star has a semimajor axis of 30 au. The dashed line represents the density distribution at time zero. It can be noted that in the region where the protoplanet is located, indicated by the grey vertical line, all simulated cases (B301, B304, B310, and B314) exhibit a gas quantity very close to the initial state of the disc, meaning that the secondary star no longer influences this region of the disc as much as before. However, in the regions closer to the edge of the disc, the available material for accretion has been reduced. The peaks in the red line are caused by the secondary star reaching pericentre at that moment, generating a strong density spiral. In this case, which represents the model with a more massive secondary star, the peaks are more pronounced. In the other semimajor axis cases, these peaks are absent because the secondary star was not at pericentre at the corresponding time.

The evolution of the semimajor axis in the secondary star at 30 au, is presented in the bottom panel of Fig. 17. In most of the cases, the protoplanets exhibit an evolution similar to the scenario without the binary (SB). In the case where the secondary star has a semimajor axis of 30 au, there is less material removal due the disc. We extended the simulation time to 7 thousand years to analyse the complete evolution up to the dissipation of the gas disc.

Within this set of cases, we can also notice the eccentricity of the protoplanets, as shown in the central panel of Fig. 17, where the peaks due to the secondary star’s pericentre are more spaced out compared to the previous cases due to the larger orbital period of the star. Case B314 is more influenced due to the star’s mass, resulting in more noticeable migration. Additionally, the peaks corresponding to the star’s approach exhibit greater variability when compared to other cases. In the B301 case, for example, short-period variations are nearly imperceptible.

The top panel of Fig. 17 shows the protoplanet’s mass evolution for a secondary star’s semimajor axis of 30 au. In this scenario, the B301 and B304 cases show mass increments closely resembling the case without the binary (SB). On the other hand, simulations B310 and B314 depict growth of over 0.4 Jupiter masses when compared to other cases.

Notably, the two simulations involving larger secondary star masses suggest that more massive companion stars influence the amount of mass accreted by the protoplanet. In this case, the greater the mass of the companion star, the larger the mass of the planet formed orbiting the primary star.

4.1 Planet accretion and disc relative velocity

The simulations conducted employed the modified fargo code, as described in Müller & Kley (2012). In essence, the code addresses the accretion of material by the planet, considering the gravitational interaction between the embedded planet and the gas disc, the formation of gaps in the disc, the dynamics of gas, as well as the temporal evolution of the system. The combination of these aspects enables the investigation of material delivery and accretion by the planet in a protoplanetary disc context (Müller & Kley 2013).

In Müller & Kley (2012), the authors introduced the disc’s eccentricity by modelling the dynamics of protoplanetary discs in binary systems with various features, including eccentricity. They employed isothermal simulations of the standard model, varying the aspect ratio |$(H/r)$|⁠, to explore the evolution of disc eccentricity.

Therefore, the implementation of disc eccentricity in the code was accomplished through the analysis of the temporal evolution of this parameter in isothermal simulations with different disc configurations. This approach allowed for an analysis of the effects of the aspect ratio on the variation of disc eccentricity and its implications for the dynamics of the studied binary system.

Disc eccentricity can influence the material accretion rate in the protoplanetary disc. In eccentric discs, the mass distribution and gas dynamics may be altered due to the eccentric orbits of gas cells around the planet.

The mass acquired by the planet in eccentric discs may vary depending on the disc eccentricity, impacting the efficiency of mass transfer from the disc to the planet and, consequently, the material acquisition rate. As we have seen, a nearby binary can induce spirals in the disc, altering its density. These variations in density near the protoplanet are crucial for the amount of material it accretes and its growth.

Models incorporating the disc and planet eccentricity are crucial for a comprehensive understanding of accretion dynamics in planetary systems. Adding the eccentricity in torque and migration calculations provides significant insights into the interaction between the planet and the disc.

Müller & Kley (2013) elucidated the relationship between disc eccentricity and disc velocity, showing that discs with varying levels of eccentricity may exhibit fluctuations in the gas velocity. The presence of eccentricity can alter the overall dynamics of the disc and influence the speed of gas motion.

The study regarded gas disc velocity as a significant parameter in accretion dynamics. Gas velocity in the protoplanetary disc influences angular momentum transfer and the formation of structures such as vortices and spirals within the disc. The correlation between gas disc velocity and accretion rate reveals that different velocity profiles can lead to distinct feeding patterns for the planet. Gas velocity in the disc can affect both the amount and the speed at which material is transferred to the planet.

The relative velocity between the nearby gas disc and the protoplanet directly affects the accretion rate of the protoplanet. Then, we verified how significantly this relative velocity changes when the secondary star is considered, compared to the case without the secondary star. We made a comparison for the binary system B101 at two different moments: one when the secondary star is located at its pericentre and another when it is located at its apocentre.

In Figs 18 and 19, respectively, we present the two components of disc velocity: radial and azimuthal. The planets are depicted as white circles, and regions of higher relative velocities are marked with white stars. The Hill radius of the protoplanet is about 0.15 au. In the plots, the x- and y-axes are in astronomical units.

In the case with the secondary star at its apocentre (Fig. 18, bottom-left plot), the region near the protoplanet shows no significant variations in relative velocity, suggesting no major implications for disc material acquisition. The white star, indicating the region of highest relative velocity.

In Fig. 18 (bottom-right plot), we show the scenario with the binary at the pericentre. Again, we found small variations in relative velocity near the protoplanet.

In the top plots of Fig. 18, we present scenarios without the binary at the same instants as the previously analysed scenario. We note that the relative velocity variation is similar to the case with the binary star. We can observe that, in all cases, the order of magnitude of the velocities is the same. The scenario without the binary star represents the ideal condition, where there is no disc eccentricity due to the presence of the secondary star, allowing for a symmetric disc, as used in Kley (1999) to calculate the disc material accretion rate by the protoplanet.

In Fig. 19, we present the azimuthal velocity of the gas disc. In this component, the cases with a binary star show more velocity variation than those without the binary, however, this variation is within the same order of magnitude.

Analysis of disc velocities near protoplanets showed that the largest relative velocity for the case with the binary is less than twice the value for the case without the binary in the vicinity of the protoplanet. Therefore, it is reasonable to adopt the same prescription for the accretion of material.

4.2 Comparison

Subsequently, we will proceed with the analysis of the systems for the cases generated in each discrete range of the secondary star’s semimajor axis. In Figs 20–22, we recap the final results for the semimajor axis, mass, and eccentricity of the protoplanet, respectively, for all presented cases at the same time, 3 thousand years.

For the semimajor axis of the protoplanets, we recall that the protoplanet was initially placed at 3 au, and, as shown in Fig. 20, the larger the semimajor axis of the secondary star, the less its influence on the protoplanet migration. The 30 au case (blue) exhibits less migration than the 20 au cases (green). However, the mass of the secondary star is also a determining factor in protoplanet migration. Scenarios where the mass of the secondary star is greater result in more significant migration of the planet toward the primary star. For the stellar companion at 10 au (purple), most of the protoplanets were ejected from the system, and the only surviving protoplanet was found when the binary star had a mass of |$0.1 {\rm M}_{\odot }$|⁠.

Fig. 21 presents the final mass values for the formed planet in all cases. For the semimajor axis of the secondary star at 10 au (purple), the only case that persists until 3 thousand years is when the secondary star has a mass of |$0.1\, {\rm M}_{\odot }$|⁠. In this scenario, the protoplanet grew until reached a mass of approximately |$1.5 M_\mathrm{ J}$|⁠. The other cases at 10 au also showed growth, but after some time, the protoplanets were either ejected from the system or collided with the primary star.

For the 20 au case (green), the planet’s mass values reached a plateau at |$2.0 M_\mathrm{ J}$| for secondary star masses above |$0.4 \,{\rm M}_{\odot }$|⁠.

In the 30 au case (blue), the protoplanet’s mass growth follows a smooth curve, increasing as the mass of the secondary star increases. As observed in this case, the mass increases until 7 thousand years when the gas mass is reduced to nearly zero.

Fig. 22 presents the final eccentricity of the planet. For the semimajor axis of the secondary star at 10 au (purple), the only case that produced a protoplanet is when the secondary star has a mass of |$0.1\, {\rm M}_{\odot }$|⁠. For the cases of |$0.4$| and |$1.4\, {\rm M}_{\odot }$|⁠, the protoplanet is ejected from the system, while in the |$1.0\, {\rm M}_{\odot }$| case, the planet collides with the primary star. In green, we present the cases at 20 au, where the values vary, peaking at approximately 0.1 for the |$0.4\, {\rm M}_{\odot }$| case. In this scenario, the eccentricities remain low in all cases.

For the 30 au case (blue), the eccentricity grows smoothly as the mass of the secondary star increases, reaching approximately 0.2 in the last value. The relationship between the values of mass and semimajor axis of the secondary stars and the final eccentricity values still requires further investigation, as the presented behaviours do not follow an apparent pattern.

The eccentric case introduces additional forces, thereby heightening the intricacy of the analysis. Secular perturbation, arising from gravitational interactions within the planetary system, exerts influence on diverse orbital elements, encompassing eccentricity, longitude of periastron, inclination, and ascending node longitude of the planets. Particularly noteworthy is the impact of a closely situated secondary star in systems, inducing perturbations on the planet (Camargo et al. 2023). Secular perturbation induces oscillations in the planet’s eccentricity within established boundaries, manifesting on time-scales significantly exceeding its orbital periods. With the dissipation of the gas disc during planetary formation, the direct perturbation from the secondary star gains prominence in comparison to the disc perturbation. Consequently, the planet’s eccentricity aligns the secular perturbation.

5 FINAL COMMENTS

In this work, we explored some limits on the formation of planets in compact binary systems. We studied systems containing two stars, a protoplanet, and a gas disc.

The study on disc truncation in the context of compact binary systems has provided valuable insights into the dynamics of gas discs and their implications for planetary formation. The gas disc’s role in planetary formation is crucial, and in the presence of a secondary star, its influence becomes even more complex.

The simulations conducted with open boundary conditions showed that the secondary star’s presence not only directly affects the protoplanet, but also induces significant disturbances in the gas disc, which can indirectly impact the planet formation process (Müller & Kley 2013).

One key observation is the reduction in the gas disc radius when a secondary star is present, especially when it has a smaller semimajor axis, which corroborates the findings of Artymowicz & Lubow (1994).

In some cases, material from the disc is launched toward the primary star, leading to an increase in the gas density near the star. This reduction in available material poses challenges for planet formation, particularly at initial semimajor axes of 3 au or less.

Despite the mass loss from the gas disc, the influx of material towards the primary star is notable, leading to an increase in gas density. This truncated disc structure becomes the starting point for the protoplanet’s material accretion process.

We must highlight the diverse conditions we explored under which giant planet formation is possible, assuming the pre-existence of a giant planet core.

For scenarios where the secondary star is at 10 au, the influence of the binary star is most pronounced, as expected. However in the case where the secondary star has a mass of |$0.1\, {\rm M}_{\odot }$| does the protoplanet remain within the system. For secondary stars with masses of |$0.4$| and |$1.4 \,{\rm M}_{\odot }$|⁠, the protoplanets are ejected from the system, while for a secondary star with a mass of |$1.0 \,{\rm M}_{\odot }$|⁠, a collision with the primary star occurs. These results suggest that in compact binary systems with a secondary star at 10 au, planet formation is challenging, and the presence of a massive secondary star compromises the stability of planets. Despite that, there is a possibility of forming planets close to the primary star when the mass of the secondary star is small, in our model, the value of this mass for the secondary star is |$0.1\, {\rm M}_{\odot }$|⁠.

In scenarios with the secondary star at 20 au, the protoplanets generally exhibit more stable behaviour, with mass growth reaching a plateau of around |$2.0 M_\mathrm{ J}$| for secondary star masses exceeding |$0.4\, {\rm M}_{\odot }$|⁠. The eccentricities of these protoplanets remain relatively low. This suggests that within the 20 au range, planet formation is more likely to occur, especially for secondary stars with masses above |$0.4\, {\rm M}_{\odot }$|⁠.

When the secondary star is placed at 30 au, the influence on the protoplanet decreases, and the final semimajor axes of the planets tend to be closer to their initial positions. The protoplanets’ mass growth in these scenarios follows a smooth curve, increasing with the mass of the secondary star. The eccentricities of these protoplanets also increase with the mass of the secondary star.

In almost all the simulations performed, the planets show Type II migration, resulting in a gap in the gas disc.

In summary, the results emphasize that the semimajor axis and mass of the secondary star have significant effects on planet formation and the subsequent evolution of protoplanets in compact binary star systems. Even under conditions considered challenging for planetary formation, it is possible to find stability in the system.

The results are non-intuitive, mainly due to the numerous forces involved in the systems. An example of this is the greater growth of planets with a secondary star of higher mass, as observed in cases with a separation of 30 au. One might expect the secondary star to hinder the growth process in this scenario; however, the results are contrary. One of the most significant findings of this study was the analysis of increased material in the disc near the primary star, which enhances planet growth through density spirals influenced by the binary star. Simulating these systems involves a wide range of parameters, indicating that factors like density, disc viscosity, and temperature could impact the results obtained.

Understanding these dynamics is essential for predicting the prevalence and characteristics of planets in such environments. Further investigations are needed to explore additional parameters and refine our understanding of planet formation in binary star systems.

For future work, we can conduct a more detailed study of the formation of the HD 41004, HD 196885, and HD 120136 systems, as each of these exhibits certain peculiarities that can be explored.

ACKNOWLEDGEMENTS

This research was supported by the Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (proc. 316991/2023-6), and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) (proc. 2016/245610). Thanks to Rafael Sfair for the assistance with computational configuration. Thanks to the CAPES, in the scope of the Program CAPES-PrInt, process no. 88887.310463/2018-00, International Cooperation Project no. 3266.

DATA AVAILABILITY

The data underlying this article will be shared on reasonable request to the corresponding author.

REFERENCES